Optimal. Leaf size=320 \[ -\frac {(b c-a d)^4 (13 a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{16/3}}+\frac {(b c-a d)^4 (13 a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{16/3}}-\frac {(b c-a d)^4 (13 a d+2 b c) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{16/3}}+\frac {d^3 x^4 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right )}{4 b^4}+\frac {d^2 x \left (-4 a^3 d^3+15 a^2 b c d^2-20 a b^2 c^2 d+10 b^3 c^3\right )}{b^5}+\frac {x (b c-a d)^5}{3 a b^5 \left (a+b x^3\right )}+\frac {d^4 x^7 (5 b c-2 a d)}{7 b^3}+\frac {d^5 x^{10}}{10 b^2} \]
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Rubi [A] time = 0.30, antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {390, 385, 200, 31, 634, 617, 204, 628} \[ \frac {d^3 x^4 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right )}{4 b^4}+\frac {d^2 x \left (15 a^2 b c d^2-4 a^3 d^3-20 a b^2 c^2 d+10 b^3 c^3\right )}{b^5}-\frac {(b c-a d)^4 (13 a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{16/3}}+\frac {(b c-a d)^4 (13 a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{16/3}}-\frac {(b c-a d)^4 (13 a d+2 b c) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{16/3}}+\frac {d^4 x^7 (5 b c-2 a d)}{7 b^3}+\frac {x (b c-a d)^5}{3 a b^5 \left (a+b x^3\right )}+\frac {d^5 x^{10}}{10 b^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 385
Rule 390
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\left (c+d x^3\right )^5}{\left (a+b x^3\right )^2} \, dx &=\int \left (\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right )}{b^5}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^3}{b^4}+\frac {d^4 (5 b c-2 a d) x^6}{b^3}+\frac {d^5 x^9}{b^2}+\frac {(b c-a d)^4 (b c+4 a d)+5 b d (b c-a d)^4 x^3}{b^5 \left (a+b x^3\right )^2}\right ) \, dx\\ &=\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^4}{4 b^4}+\frac {d^4 (5 b c-2 a d) x^7}{7 b^3}+\frac {d^5 x^{10}}{10 b^2}+\frac {\int \frac {(b c-a d)^4 (b c+4 a d)+5 b d (b c-a d)^4 x^3}{\left (a+b x^3\right )^2} \, dx}{b^5}\\ &=\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^4}{4 b^4}+\frac {d^4 (5 b c-2 a d) x^7}{7 b^3}+\frac {d^5 x^{10}}{10 b^2}+\frac {(b c-a d)^5 x}{3 a b^5 \left (a+b x^3\right )}+\frac {\left ((b c-a d)^4 (2 b c+13 a d)\right ) \int \frac {1}{a+b x^3} \, dx}{3 a b^5}\\ &=\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^4}{4 b^4}+\frac {d^4 (5 b c-2 a d) x^7}{7 b^3}+\frac {d^5 x^{10}}{10 b^2}+\frac {(b c-a d)^5 x}{3 a b^5 \left (a+b x^3\right )}+\frac {\left ((b c-a d)^4 (2 b c+13 a d)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{5/3} b^5}+\frac {\left ((b c-a d)^4 (2 b c+13 a d)\right ) \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{5/3} b^5}\\ &=\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^4}{4 b^4}+\frac {d^4 (5 b c-2 a d) x^7}{7 b^3}+\frac {d^5 x^{10}}{10 b^2}+\frac {(b c-a d)^5 x}{3 a b^5 \left (a+b x^3\right )}+\frac {(b c-a d)^4 (2 b c+13 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{16/3}}-\frac {\left ((b c-a d)^4 (2 b c+13 a d)\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{5/3} b^{16/3}}+\frac {\left ((b c-a d)^4 (2 b c+13 a d)\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{4/3} b^5}\\ &=\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^4}{4 b^4}+\frac {d^4 (5 b c-2 a d) x^7}{7 b^3}+\frac {d^5 x^{10}}{10 b^2}+\frac {(b c-a d)^5 x}{3 a b^5 \left (a+b x^3\right )}+\frac {(b c-a d)^4 (2 b c+13 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{16/3}}-\frac {(b c-a d)^4 (2 b c+13 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{16/3}}+\frac {\left ((b c-a d)^4 (2 b c+13 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{5/3} b^{16/3}}\\ &=\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^4}{4 b^4}+\frac {d^4 (5 b c-2 a d) x^7}{7 b^3}+\frac {d^5 x^{10}}{10 b^2}+\frac {(b c-a d)^5 x}{3 a b^5 \left (a+b x^3\right )}-\frac {(b c-a d)^4 (2 b c+13 a d) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{16/3}}+\frac {(b c-a d)^4 (2 b c+13 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{16/3}}-\frac {(b c-a d)^4 (2 b c+13 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{16/3}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 313, normalized size = 0.98 \[ \frac {-\frac {70 (b c-a d)^4 (13 a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+\frac {140 (b c-a d)^4 (13 a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}+\frac {140 \sqrt {3} (b c-a d)^4 (13 a d+2 b c) \tan ^{-1}\left (\frac {2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3}}+315 b^{4/3} d^3 x^4 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right )+1260 \sqrt [3]{b} d^2 x \left (-4 a^3 d^3+15 a^2 b c d^2-20 a b^2 c^2 d+10 b^3 c^3\right )+180 b^{7/3} d^4 x^7 (5 b c-2 a d)+\frac {420 \sqrt [3]{b} x (b c-a d)^5}{a \left (a+b x^3\right )}+126 b^{10/3} d^5 x^{10}}{1260 b^{16/3}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 1619, normalized size = 5.06 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 529, normalized size = 1.65 \[ -\frac {\sqrt {3} {\left (2 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d - 40 \, a^{2} b^{3} c^{3} d^{2} + 70 \, a^{3} b^{2} c^{2} d^{3} - 50 \, a^{4} b c d^{4} + 13 \, a^{5} d^{5}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{4}} - \frac {{\left (2 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d - 40 \, a^{2} b^{3} c^{3} d^{2} + 70 \, a^{3} b^{2} c^{2} d^{3} - 50 \, a^{4} b c d^{4} + 13 \, a^{5} d^{5}\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{4}} - \frac {{\left (2 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d - 40 \, a^{2} b^{3} c^{3} d^{2} + 70 \, a^{3} b^{2} c^{2} d^{3} - 50 \, a^{4} b c d^{4} + 13 \, a^{5} d^{5}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2} b^{5}} + \frac {b^{5} c^{5} x - 5 \, a b^{4} c^{4} d x + 10 \, a^{2} b^{3} c^{3} d^{2} x - 10 \, a^{3} b^{2} c^{2} d^{3} x + 5 \, a^{4} b c d^{4} x - a^{5} d^{5} x}{3 \, {\left (b x^{3} + a\right )} a b^{5}} + \frac {14 \, b^{18} d^{5} x^{10} + 100 \, b^{18} c d^{4} x^{7} - 40 \, a b^{17} d^{5} x^{7} + 350 \, b^{18} c^{2} d^{3} x^{4} - 350 \, a b^{17} c d^{4} x^{4} + 105 \, a^{2} b^{16} d^{5} x^{4} + 1400 \, b^{18} c^{3} d^{2} x - 2800 \, a b^{17} c^{2} d^{3} x + 2100 \, a^{2} b^{16} c d^{4} x - 560 \, a^{3} b^{15} d^{5} x}{140 \, b^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 905, normalized size = 2.83 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 509, normalized size = 1.59 \[ \frac {{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} x}{3 \, {\left (a b^{6} x^{3} + a^{2} b^{5}\right )}} + \frac {14 \, b^{3} d^{5} x^{10} + 20 \, {\left (5 \, b^{3} c d^{4} - 2 \, a b^{2} d^{5}\right )} x^{7} + 35 \, {\left (10 \, b^{3} c^{2} d^{3} - 10 \, a b^{2} c d^{4} + 3 \, a^{2} b d^{5}\right )} x^{4} + 140 \, {\left (10 \, b^{3} c^{3} d^{2} - 20 \, a b^{2} c^{2} d^{3} + 15 \, a^{2} b c d^{4} - 4 \, a^{3} d^{5}\right )} x}{140 \, b^{5}} + \frac {\sqrt {3} {\left (2 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d - 40 \, a^{2} b^{3} c^{3} d^{2} + 70 \, a^{3} b^{2} c^{2} d^{3} - 50 \, a^{4} b c d^{4} + 13 \, a^{5} d^{5}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d - 40 \, a^{2} b^{3} c^{3} d^{2} + 70 \, a^{3} b^{2} c^{2} d^{3} - 50 \, a^{4} b c d^{4} + 13 \, a^{5} d^{5}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d - 40 \, a^{2} b^{3} c^{3} d^{2} + 70 \, a^{3} b^{2} c^{2} d^{3} - 50 \, a^{4} b c d^{4} + 13 \, a^{5} d^{5}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 416, normalized size = 1.30 \[ x\,\left (\frac {10\,c^3\,d^2}{b^2}-\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {2\,a\,d^5}{b^3}-\frac {5\,c\,d^4}{b^2}\right )}{b}-\frac {a^2\,d^5}{b^4}+\frac {10\,c^2\,d^3}{b^2}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^5}{b^3}-\frac {5\,c\,d^4}{b^2}\right )}{b^2}\right )-x^7\,\left (\frac {2\,a\,d^5}{7\,b^3}-\frac {5\,c\,d^4}{7\,b^2}\right )+x^4\,\left (\frac {a\,\left (\frac {2\,a\,d^5}{b^3}-\frac {5\,c\,d^4}{b^2}\right )}{2\,b}-\frac {a^2\,d^5}{4\,b^4}+\frac {5\,c^2\,d^3}{2\,b^2}\right )+\frac {d^5\,x^{10}}{10\,b^2}-\frac {x\,\left (a^5\,d^5-5\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3-10\,a^2\,b^3\,c^3\,d^2+5\,a\,b^4\,c^4\,d-b^5\,c^5\right )}{3\,a\,\left (b^6\,x^3+a\,b^5\right )}+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,{\left (a\,d-b\,c\right )}^4\,\left (13\,a\,d+2\,b\,c\right )}{9\,a^{5/3}\,b^{16/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^4\,\left (13\,a\,d+2\,b\,c\right )}{9\,a^{5/3}\,b^{16/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^4\,\left (13\,a\,d+2\,b\,c\right )}{9\,a^{5/3}\,b^{16/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.43, size = 546, normalized size = 1.71 \[ x^{7} \left (- \frac {2 a d^{5}}{7 b^{3}} + \frac {5 c d^{4}}{7 b^{2}}\right ) + x^{4} \left (\frac {3 a^{2} d^{5}}{4 b^{4}} - \frac {5 a c d^{4}}{2 b^{3}} + \frac {5 c^{2} d^{3}}{2 b^{2}}\right ) + x \left (- \frac {4 a^{3} d^{5}}{b^{5}} + \frac {15 a^{2} c d^{4}}{b^{4}} - \frac {20 a c^{2} d^{3}}{b^{3}} + \frac {10 c^{3} d^{2}}{b^{2}}\right ) + \frac {x \left (- a^{5} d^{5} + 5 a^{4} b c d^{4} - 10 a^{3} b^{2} c^{2} d^{3} + 10 a^{2} b^{3} c^{3} d^{2} - 5 a b^{4} c^{4} d + b^{5} c^{5}\right )}{3 a^{2} b^{5} + 3 a b^{6} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} a^{5} b^{16} - 2197 a^{15} d^{15} + 25350 a^{14} b c d^{14} - 132990 a^{13} b^{2} c^{2} d^{13} + 418280 a^{12} b^{3} c^{3} d^{12} - 874635 a^{11} b^{4} c^{4} d^{11} + 1271886 a^{10} b^{5} c^{5} d^{10} - 1302400 a^{9} b^{6} c^{6} d^{9} + 922680 a^{8} b^{7} c^{7} d^{8} - 422235 a^{7} b^{8} c^{8} d^{7} + 97570 a^{6} b^{9} c^{9} d^{6} + 7194 a^{5} b^{10} c^{10} d^{5} - 10200 a^{4} b^{11} c^{11} d^{4} + 1435 a^{3} b^{12} c^{12} d^{3} + 330 a^{2} b^{13} c^{13} d^{2} - 60 a b^{14} c^{14} d - 8 b^{15} c^{15}, \left (t \mapsto t \log {\left (\frac {9 t a^{2} b^{5}}{13 a^{5} d^{5} - 50 a^{4} b c d^{4} + 70 a^{3} b^{2} c^{2} d^{3} - 40 a^{2} b^{3} c^{3} d^{2} + 5 a b^{4} c^{4} d + 2 b^{5} c^{5}} + x \right )} \right )\right )} + \frac {d^{5} x^{10}}{10 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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